With the newly elected Congress more closely divided than at any time since Dwight D. Eisenhower was president, members of both parties are wrestling with how to run a branch of government where any question -- from the confirmation of cabinet members to assigning committee office space -- could create deadlock.

There is an uncertain mixture of expectations, and Senator John McCain, the maverick Republican from Arizona, conveyed the range of them. He said: ''The likely scenario is gridlock. The other scenario is that members of both parties figure out that the American people are fed up. They don't get it -- why we can't work together on issues that are important to them and their families.''

Even those steeped in the history of Congress say there has never been a situation quite like what Washington is about to face. In 1953, a very popular president, General Eisenhower, dealt with an 83rd Congress controlled by Republicans by only eight votes in the House and two in the Senate.

But those were easier times, when cooperation across party lines was instinctive on many issues, not just one tactical alternative as it is today. The new president, whoever he is, will take office with the most tenuous of electoral margins, considerable public doubt that he won fairly, and a severe level of rancor on Capitol Hill.

''The partisanship is extremely high, worse than at any time since the Vietnam War,'' said Wendy J. Schiller, associate professor of political science at Brown University. She said the lack of trust that leaders showed in one another today matched the bitterness their predecessors displayed in the Depression.

In the Senate, which will be divided 50-50 if Gov. George W. Bush of Texas becomes president and Joseph I. Lieberman stays in his seat as senator from Connecticut, Democrats are demanding co-chairmanships of committees and the right to propose legislation on the floor without parliamentary obstacles.

In the House, where the elections reduced the Republican majority to 9 votes from 13, the Republican leadership can expect to face difficulties not only with the Democrats but from senior Republicans who will have to be bypassed when the leaders choose from among many applicants for committee chairmanships. Moreover, Democrats are asking for more committee seats and procedures that will make it easier for them to offer amendments. It will be up to Republicans to decide whether to conciliate them.

There is talk of compromises. Senator John B. Breaux of Louisiana, a middle-of-the-road Democrat who has tried to form alliances with Republicans on health care, said: ''I think bipartisanship instead of being just a political theory is going to be a political necessity. There is an enormous opportunity for those in the center to build coalitions.''

And various members suggested topics for compromise were plentiful, mostly leftovers from the departing Congress like specific tax cuts, for example the inheritance tax; a patients' bill of rights with a limited right to sue health maintenance organizations; and possibly some kind of a prescription drug plan under Medicare.

But there are doubts, too.

Representative John D. Dingell, the Michigan Democrat and the dean of the House, said Democrats remained angry over how they had been treated since Republicans took the House in 1994 -- ''a wide array of affronts that go back at least six years, a wide array of concerns about basic unfairness, exclusion from the legislative process.''

Senator Thad Cochran, a Mississippi Republican widely respected among Democrats, said of them, ''I think they are going to be hard to get along with.''

The first place where members of Congress have to work together is in the House Republican Conference, or caucus. When Republicans took over the House, they imposed six-year term limits on chairmen of committees and subcommittees. Thirteen or 14 full House committees will get new chairmen, and in only a handful is there no competition.

So next week 26 senior House Republicans will present themselves to the party's steering committee, headed by Speaker J. Dennis Hastert of Illinois, to apply for the vacant jobs. Before they take effect, the steering committee's selections must be ratified by the Republican conference just as the new Congress convenes.

The touchiest case may be the Banking Committee. Representative Marge Roukema, a New Jersey moderate who sometimes bucks the leadership and the only woman in contention for a committee chairmanship, has the seniority but is opposed by Representative Richard H. Baker of Louisiana, who is more conservative.

Mr. Hastert has not specified how the steering committee plans to choose. One friend of his, the former minority leader Bob Michel of Illinois, said he had urged Mr. Hastert to ''honor the seniority system.''

''Otherwise you're going to open up a whole can of worms,'' Mr. Michel said. ''You've already got enough troubles.''

But since Speaker Newt Gingrich bypassed seniority in picking a couple of chairmen in 1995, other considerations have come into play. Representative Ray LaHood, another Illinois Republican, said he felt Mr. Hastert would stress ''loyalty to leadership'' as one qualification, along with the ability to work with others, including Democrats. ''We've got to be able to reach out to centrist Democrats, conservative Democrats,'' Mr. LaHood said.

But duels -- or deals -- between the parties may come to a head more quickly in the Senate, whether the split is 50-50 or 51-49, as would be the case if Mr. Lieberman becomes vice president and his seat is filled by an appointed Republican.

Senator Tom Daschle of South Dakota, the Democratic leader, argues that in either case, power should be shared. ''Any time you have a margin that close it requires that both parties have an interest in legislation,'' Mr. Daschle said.

Most Republicans are hostile to that idea. ''There's only one majority and one minority in the Senate and the majority runs the committees,'' said Senator Robert C. Smith of New Hampshire, who has worked well with Democrats on the Environment and Public Works Committee.

And Senator Cochran said he believed Senator Trent Lott of Mississippi, the Republican leader, thought he could not get Republicans to accept less than a one-vote majority on most committees. This year, each of the Senate's 17 legislative committees has a two-vote Republican majority.

Not every Republican chairman feels that way. Senator Richard G. Lugar of Indiana, chairman of the Agriculture Committee, asked if he could accept an even party split on his committee, said, ''I suppose so.'' He noted that the committee rarely reports bills without at least a two-thirds vote.

Mr. McCain, chairman of the Commerce Committee, predicted, ''We're going to have to have 50-50 committees.'' He said he had told his committee's senior Democrat, Senator Ernest F. Hollings of South Carolina, that ''I am fully ready to accept an even committee'' while retaining the chairmanship himself.

''We were shutting out the Democrats from their legitimate rights as a minority in the Senate, and there is a lot of anger among the Democrats about it,'' he said. ''We have to do business differently.''

Equal numbers, with a Republican chairman, would be one obvious compromise between Mr. Daschle's proposal of co-chairmen and Mr. Lott's belief in Republican majorities. Assuring the Democrats more than the current one-third of the budgets and office space will be easier.

There is particular reason for a speedy decision. Some of the Senate committees that consider high-level cabinet appointments, especially Foreign Relations and Judiciary, have already become evenly divided because of retirements and the defeats of incumbents.

And the Finance Committee, which would consider a potential secretary of the treasury, may not be able to meet at all without a Lott-Daschle agreement because its chairman, Senator William V. Roth Jr. of Delaware, was defeated. A new chairman can be approved only by the Senate, and that action is subject to a filibuster.

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One early legislative test, no matter who is president, will probably come over campaign finance legislation. Senator McCain and Russell D. Feingold, Democrat of Wisconsin, say they believe they picked up enough votes through the elections to pass the measure banning unregulated soft money contributions.

''We think that this filibuster has had it,'' Mr. Feingold said. ''We intend to have this come up right away. That means January to me.''

Mr. Lugar said he was hopeful for a productive session, ''but if both parties wish to hogtie each other, then we won't do much else for a while.''

Representative Steny H. Hoyer, Democrat of Maryland, said the tensions over the presidential election and the efforts by Representative Tom DeLay of Texas, the Republican whip, to help Mr. Bush would make cooperation very difficult unless whoever is president leans hard on his party to cooperate with the other side.

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A version of this article appears in print on December 3, 2000, on Page 1001001 of the National edition with the headline: CONTESTING THE VOTE: CONGRESS; Sharply Split Congress Grapples With How to Keep Things Going. Order Reprints|Today's Paper|Subscribe